Equation of Motion with Steering Control

Differential steering of a single-axle vehicle in planar turning motion. Velocities in a body-fixed frame, including wheel angular velocities and yaw rate. Position and velocity relationships in the global reference frame. MATLAB programming for a differentially-driven single-axle vehicle with the center of gravity on the axle.

• Kinematics
• Vehicle Dynamics
• Steering Control
• Differential Steering
• MATLAB Programming

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1. Equation of Motion with Steering Control ME5670 Lecture 3 Thomas Gillespie, Fundamentals of Vehicle Dynamics , SAE, 1992. http://www.me.utexas.edu/~longoria/VSDC/clog.html http://www.slideshare.net/NirbhayAgarwal/four-wheel-steering-system Class timing Monday: 14:30 Hrs 16:00 Hrs Thursday: 16:30 Hrs 17:30 Hrs Date: 19/01/2015

2. Kinematic Models of 2D Steering Turning Example: Differential steering of a single-axle vehicle in planar, turning motion For the simple vehicle model shown to the left, there are negligible forces at point A. This could be a pivot, caster, or some other omni-directional type wheel. Assume the vehicle has constant forward velocity, U. Assume the wheels roll without slip and cannot slip laterally. Designate the right wheel 1 and the left 2 . What are the velocities in a body-fixed frame? Also find the yaw angular rate.

3. Solution 1. Apply 2. Velocity at the left wheel Applying the lateral constraint ?2?= 0 2. Velocity at the right wheel 3. Velocity of CG: where, ?1 and ?2 are wheel angular velocity and ?? is the wheel radius 5. Yaw rate:

4. Kinematics: Example 2 Position and velocity in inertial frame Vehicle kinematic state in the inertial frame . Velocities in the local reference frame are related with the inertial frame by the rotation matrix . Velocities in the global reference frame From Example 1, we have Velocities in the global reference frame in terms of wheel velocities are

5. Kinematics: Example 3 Differentially-driven single axle vehicle with CG on axle For a kinematic model for a vehicle with CG on axle and Velocities in the global reference frame MATLAB programming clear all global R_w B omegaw1 omegaw2 R_w = 0.05; B = 0.18; omegaw1 = 4; omegaw2 = 2; Xi0=[0,0,0]; [t,Xi] = ode45(@DS_vehicle,[0 10],Xi0); N = length(t); figure(1) plot(Xi(:,1),Xi(:,2)), axis([-1.0 1.0 -0.5 1.5]), axis('square') xlabel('X'), ylabel('Y') function Xidot = DS_vehicle(t,Xi) global R_w B omegaw1 omegaw2 X = Xi(1); Y = Xi(2); psi = Xi(3); Xdot = 0.5*cos(psi)*R_w*(omegaw1+omegaw2); Ydot = 0.5*sin(psi)*R_w*(omegaw1+omegaw2); psidot = R_w*(omegaw1-omegaw2)/B; Xidot=[Xdot;Ydot;psidot]; Courtesy: Prof. R.G. Longoria

6. Kinematics: 2D Animation Differentially-driven single axle vehicle with CG on axle Vehicle State function [xb, yb, xfc, yfc, xrlw, yrlw, xrrw, yrrw] = vehicle_state(q,u) global L B R_w % x and y are the coordinates at the rear axle center - CG location % u is a control input x = q(1); y = q(2); psi = q(3); % xfc and yfc are coordinates of a center pivot at front % then the pivot point is located w.r.t CG at a distance L % Find coordinates to draw wheels % rear left wheel xrlwf = xrl + R_w*cos(psi); yrlwf = yrl + R_w*sin(psi); xrlwr = xrl - R_w*cos(psi); yrlwr = yrl - R_w*sin(psi); % rear right wheel xrrwf = xrr + R_w*cos(psi); yrrwf = yrr + R_w*sin(psi); xrrwr = xrr - R_w*cos(psi); yrrwr = yrr - R_w*sin(psi); xrlw = [xrlwf, xrlwr]; yrlw = [yrlwf, yrlwr]; xrrw = [xrrwf, xrrwr]; yrrw = [yrrwf, yrrwr]; xfc = x + 1*L*cos(psi); yfc = y + 1*L*sin(psi); % Find coordinates of vehicle base xfl = xfc - 0.5*B*sin(psi); yfl = yfc + 0.5*B*cos(psi); xfr = xfc + 0.5*B*sin(psi); yfr = yfc - 0.5*B*cos(psi); xrl = x - 0.5*B*sin(psi); yrl = y + 0.5*B*cos(psi); xrr = x + 0.5*B*sin(psi); yrr = y - 0.5*B*cos(psi); xb = [xfl, xfr, xrr, xrl, xfl]; % x coordinates for vehicle base yb = [yfl, yfr, yrr, yrl, yfl]; % y coordinates for vehicle base Courtesy: Prof. R.G. Longoria

7. Kinematics: 2D Animation Differentially-driven single axle vehicle with CG on axle Rk4 Solver function [T,X]=rk4fixed(Fcn,Tspan,X0,N) h = (Tspan(2)-Tspan(1))/N; halfh = 0.5*h; neqs=size(X0); X=zeros(neqs(1),N); T=zeros(1,N); X(:,1)=X0; T(1)=Tspan(1); Td = Tspan(1); Xd = X0; for i=2:N, RK1 = feval(Fcn,Td,Xd); Thalf = Td + halfh; Xtemp = Xd + halfh*RK1; RK2 = feval(Fcn,Thalf,Xtemp); Xtemp = Xd + halfh*RK2; RK3 = feval(Fcn,Thalf,Xtemp); Tfull = Td + h; Xtemp = Xd + h*RK3; RK4 = feval(Fcn,Tfull,Xtemp); X(:,i) = Xd + h*(RK1+2.0*(RK2+RK3)+RK4)/6; T(i) = Tfull; Xd = X(:,i); Td = T(i); end X=X';T=T'; Courtesy: Prof. R.G. Longoria

8. 2D Animation Sim_2Danim.m % Vehicle_State provides spatial state information for the vehicle clear all; % Clear all variables close all; % Close all figures global L B R_w omegaw1 omegaw2 % Geometric vehicle parameters L = 0.20; % wheel base B = 0.18; % rear axle track width R_w = 0.05; % wheel radius % Initial location and orientation of the vehicle CG x0 = 0; y0 = 0; psi0 = 0*pi/180; % psi = yaw angle in radians fig1 = figure(1); axis([-1.0 1.0 -0.5 1.5]); axis('square') xlabel('X'), ylabel('Y') hold on; q0=[x0,y0,psi0]; [xb, yb, xfc, yfc, xrlw, yrlw, xrrw, yrrw] = vehicle_state(q0,0); % Plot vehicle and define component plots plotzb = plot(xb, yb); % Plot robot base plotzfc = plot(xfc, yfc, 'o'); % Plot front pivot plotzrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel plotzrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel % Set handle graphics parameters and plotting modes set(gca, 'drawmode','fast'); set(plotzb, 'erasemode', 'xor'); % use 'xor' rather than 'none' to redraw set(plotzfc, 'erasemode', 'xor'); set(plotzrlw, 'erasemode', 'xor'); set(plotzrrw, 'erasemode', 'xor'); q1 = q0; % Set initial state to q1 for simulation % Fixed wheel speed command - should make a circle! omegaw1 = 2; omegaw2 = 1; Courtesy: Prof. R.G. Longoria

9. Kinematics: 2D Animation % Parameters related to simulations tfinal = 100; N = 30; % Number of iterations dt = tfinal/N; % Time step interval t = [0:dt:tfinal]; % Beginning of simulation and animation for i = 1:N+1 to = t(i); tf = t(i)+dt; % integrate from to to tf [t2,q2]=rk4fixed('dssakv',[to tf],q1',2); t1 = t2(2); % keep only the last point q1 = q2(2,:); % store q2 in q1 for next step % capture the state of the vehicle for animation [xb, yb, xfc, yfc, xrlw, yrlw, xrrw, yrrw] = vehicle_state(q1, 0); plot(xfc,yfc,'r.') % Plot vehicle - updates data in each plot set(plotzb,'xdata',xb); set(plotzb,'ydata',yb); set(plotzfc,'xdata',xfc); set(plotzfc,'ydata',yfc); set(plotzrlw,'xdata',xrlw); set(plotzrlw,'ydata',yrlw); set(plotzrrw,'xdata',xrrw); set(plotzrrw,'ydata',yrrw); pause(0.2); % Pause by X seconds for slower animation end Courtesy: Prof. R.G. Longoria

10. Ackerman Steering : A Tricycle Single-axle vehicle with front-steered wheel; rolling rear wheels For a given steer angle ? and C.G. velocity along x as v. Velocities in the inertial frame is given by The input control variables are ? = ??? and steer angle ? Here, C.G. is located at the rear axle, its velocity is given by ? =1 2?1+ ?2, where ?1= ???1 and ?2= ???2 Forward velocty at the front wheel is ? along x. ? . Because of steering, the velocity along the path of the wheel is ??= The velocity lateral to the wheel is ??= ??sin ? = ? tan(?) ??? cos(?) ? =? ??? Therefore, ??= ?tan ? = tan ? = Courtesy: Prof. R.G. Longoria

11. Kinematics: 2D Animation Differentially-driven single axle tricycle % Find coordinates to draw wheels % rear left wheel xrlwf = xrl + R_w*cos(psi); yrlwf = yrl + R_w*sin(psi); xrlwr = xrl - R_w*cos(psi); yrlwr = yrl - R_w*sin(psi); % rear right wheel xrrwf = xrr + R_w*cos(psi); yrrwf = yrr + R_w*sin(psi); xrrwr = xrr - R_w*cos(psi); yrrwr = yrr - R_w*sin(psi); % define the states % front center point (not returned) qfc = [xfc, yfc]; % body x-y points xb = [xfl, xfr, xrr, xrl, xfl]; yb = [yfl, yfr, yrr, yrl, yfl]; % front wheel x-y points xfw = [xfwf, xfwr]; yfw = [yfwf, yfwr]; % rear-left wheel x-y points xrlw = [xrlwf, xrlwr]; yrlw = [yrlwf, yrlwr]; % rear-right wheel x-y points xrrw = [xrrwf, xrrwr]; yrrw = [yrrwf, yrrwr]; Tricycle State function [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = tricycle_state(q, u) global L B R_w % x and y are the coordinates at the rear axle center - CG location % u is a control input x = q(1); y = q(2); psi = q(3); v = u(1); delta = u(2); % xfc and yfc are coordinates of a center pivot at front % then the pivot point is located w.r.t CG at a distance L xfc = x + L*cos(psi); yfc = y + L*sin(psi); % Find coordinates of vehicle base xfl = xfc - 0.5*B*sin(psi); yfl = yfc + 0.5*B*cos(psi); xfr = xfc + 0.5*B*sin(psi); yfr = yfc - 0.5*B*cos(psi); xrl = x - 0.5*B*sin(psi); yrl = y + 0.5*B*cos(psi); xrr = x + 0.5*B*sin(psi); yrr = y - 0.5*B*cos(psi); % end points of the front-steered wheel xfwf = xfc + R_w*cos(psi+delta); yfwf = yfc + R_w*sin(psi+delta); xfwr = xfc - R_w*cos(psi+delta); yfwr = yfc - R_w*sin(psi+delta); Courtesy: Prof. R.G. Longoria

12. 2D Animation % sim_tricycle_model.m clear all; % Clear all variables close all; % Close all figures global L B R_w vc delta_radc function qdot = ks_tricycle_kv(t,q) global L vc delta_radc delta_max_deg R_w % Physical parameters of the tricycle L = 2.040; %0.25; % [m] B = 1.164; %0.18; % Distance between the rear wheels [m] m_max_rpm = 8000; % Motor max speed [rpm] gratio = 20; % Gear ratio R_w = 13/39.37; % Radius of wheel [m] % L is length between the front wheel axis and rear wheel %axis [m] % vc is speed command % delta_radc is the steering angle command % State variables x = q(1); y = q(2); psi = q(3); % Control variables v = vc; delta = delta_radc; % kinematic model xdot = v*cos(psi); ydot = v*sin(psi); psidot = v*tan(delta)/L; qdot = [xdot;ydot;psidot]; % Parameters related to vehicle m_max_rads = m_max_rpm*2*pi/60; % Motor max speed [rad/s] w_max_rads = m_max_rads/gratio; % Wheel max speed [rad/s] v_max = w_max_rads*R_w; % Max robot speed [m/s] % Initial values x0 = 0; % Initial x coodinate [m] y0 = 0; % Initial y coodinate [m] psi_deg0 = 0; % Initial orientation of the robot (theta [deg]) % desired turn radius R_turn = 3*L; delta_max_rad = L/R_turn; % Maximum steering angle [deg] % Parameters related to simulations t_max = 10; % Simulation time [s] n = 100; % Number of iterations dt = t_max/n; % Time step interval t = [0:dt:t_max]; % Time vector (n+1 components) Courtesy: Prof. R.G. Longoria

13. 2D Animation % velocity and steering commands (open loop) v = v_max*ones(1,n+1); % Velocity vector (n+1 components) delta_rad = delta_max_rad*ones(1,n+1); % Steering angle vector (n+1 %components) [rad] psi_rad0 = psi_deg0*pi/180; % Initial orientation [rad] v0 = v(1); % Initial velocity [m/s] delta_rad0 = delta_rad(1); % Initial steering angle [rad] q0 = [x0, y0, psi_rad0]; % Initial state vector u0 = [v0, delta_rad0]; % Initial control vector fig1 = figure(1); % Figure set-up (fig1) axis([-R_turn R_turn -0*R_turn 2*R_turn]); axis('square') hold on; % Acquire the configuration of robot for plot [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = tricycle_state(q0, u0); plotqb = plot(xb, yb); % Plot vehicle base plotqfw = plot(xfw, yfw, 'r'); % Plot front wheel plotqrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel plotqrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel % Draw fast and erase fast set(gca, 'drawmode','fast'); set(plotqb, 'erasemode', 'xor'); set(plotqfw, 'erasemode', 'xor'); set(plotqrlw, 'erasemode', 'xor'); set(plotqrrw, 'erasemode', 'xor'); % Beginning of simulation for i = 1:n+1 v(i) = v_max*cos(2*delta_rad(i)); u = [v(i), delta_rad(i)]; % Set control input vc = u(1); delta_radc = u(2); to = t(i); tf = t(i)+dt; [t2,q2]=rk4fixed('ks_tricycle_kv',[to tf],q1',2); t1 = t2(2); q1 = q2(2,:); % Acquire the configuration of vehicle for plot [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = tricycle_state(q1, u); % Plot vehicle set(plotqb,'xdata',xb); set(plotqb,'ydata',yb); set(plotqfw,'xdata',xfw); set(plotqfw,'ydata',yfw); set(plotqrlw,'xdata',xrlw); set(plotqrlw,'ydata',yrlw); set(plotqrrw,'xdata',xrrw); set(plotqrrw,'ydata',yrrw); % drawnow pause(0.1); % Pause by 0.2s for slower simulation end q1 = q0; % Set initial state to z1 for simulation Courtesy: Prof. R.G. Longoria

14. 2D Animation % Plot the resultant velocity and steering angle configurations fig2 = figure(2); % Figure set-up (fig2) subplot(2,1,1); % Upper half of fig1 plot(t, v); % Plot velocity-time curve xlabel('Time [s]'); ylabel('Velocity [m/s]'); subplot(2,1,2); % Lower half of fig1 deltad = delta_rad*180/pi; % Steering angle vector (n+1 comp.) [deg] plot(t,deltad); % Plot steering angle-time curve xlabel('Time [s]'); ylabel('Steering angle [deg]'); 12 Velocity [m/s] 11 10 9 0 1 2 3 4 5 6 7 8 9 10 Time [s] 21 Steering angle [deg] 20 19 18 0 1 2 3 4 5 6 7 8 9 10 Time [s] Courtesy: Prof. R.G. Longoria

15. Kinematics: Lane Change Problem Differentially-driven Double axle Four Wheel Vehicle Four Wheel Vehicle State % end points of the front-steered wheel xfwfr = xfl + R_w*cos(psi+delta); yfwfr = yfl + R_w*sin(psi+delta); xfwrr = xfl - R_w*cos(psi+delta); yfwrr = yfl - R_w*sin(psi+delta); function [xb, yb, xfc, yfc, xfwr, yfwr,xfwl, yfwl, xrlw, yrlw, xrrw, yrrw] = Fourwheel_state(q, u) global L B R_w % end points of the front-steered wheel xfwfl = xfr + R_w*cos(psi+delta); yfwfl = yfr + R_w*sin(psi+delta); xfwrl = xfr - R_w*cos(psi+delta); yfwrl = yfr - R_w*sin(psi+delta); x = q(1); y = q(2); psi = q(3); v = u(1); delta = u(2); % locates front-center point xfc = x + L*cos(psi); yfc = y + L*sin(psi); % end points of the rear-left wheel xrlwf = xrl + R_w*cos(psi); yrlwf = yrl + R_w*sin(psi); xrlwr = xrl - R_w*cos(psi); yrlwr = yrl - R_w*sin(psi); % locates four corners xfl = xfc - 0.5*B*sin(psi); yfl = yfc + 0.5*B*cos(psi); xfr = xfc + 0.5*B*sin(psi); yfr = yfc - 0.5*B*cos(psi); xrl = x - 0.5*B*sin(psi); yrl = y + 0.5*B*cos(psi); xrr = x + 0.5*B*sin(psi); yrr = y - 0.5*B*cos(psi); % end points of the rear-right wheel xrrwf = xrr + R_w*cos(psi); yrrwf = yrr + R_w*sin(psi); xrrwr = xrr - R_w*cos(psi); yrrwr = yrr - R_w*sin(psi); ];

16. 2D Animation % sim_tricycle_model.m clear all; % Clear all variables close all; % Close all figures global L B R_w vc delta_radc % define the states % front center point (not returned) qfc = [xfc, yfc]; % Physical parameters of the tricycle L = 2.040; %0.25; % [m] B = 1.164; %0.18; % Distance between the rear wheels [m] m_max_rpm = 8000; % Motor max speed [rpm] gratio = 20; % Gear ratio R_w = 13/39.37; % Radius of wheel [m] % body x-y points xb = [xfl, xfr, xrr, xrl, xfl]; yb = [yfl, yfr, yrr, yrl, yfl]; % Parameters related to vehicle m_max_rads = m_max_rpm*2*pi/60; % Motor max speed [rad/s] w_max_rads = m_max_rads/gratio; % Wheel max speed [rad/s] v_max = w_max_rads*R_w; % Max robot speed [m/s] % left front wheel x-y points xfwl = [xfwfl, xfwrl]; yfwl = [yfwfl, yfwrl]; % Right front wheel x-y points xfwr = [xfwfr, xfwrr]; yfwr = [yfwfr, yfwrr]; % Initial values x0 = 0; % Initial x coodinate [m] y0 = 0; % Initial y coodinate [m] psi_deg0 = 0; % Initial orientation of the robot (theta [deg]) % rear-left wheel x-y points xrlw = [xrlwf, xrlwr]; yrlw = [yrlwf, yrlwr]; % desired turn radius R_turn = 3*L; delta_max_rad = L/R_turn; % Maximum steering angle [deg] % Parameters related to simulations t_max = 10; % Simulation time [s] n = 100; % Number of iterations dt = t_max/n; % Time step interval t = [0:dt:t_max]; % Time vector (n+1 components) % rear-right wheel x-y points xrrw = [xrrwf, xrrwr]; yrrw = [yrrwf, yrrwr];

17. 2D Animation % velocity and steering commands (open loop) v = v_max*ones(1,n+1); % Velocity vector (n+1 components) delta_rad = delta_max_rad*ones(1,n+1); % Steering angle vector (n+1 %components) [rad] psi_rad0 = psi_deg0*pi/180; % Initial orientation [rad] v0 = v(1); % Initial velocity [m/s] delta_rad0 = delta_rad(1); % Initial steering angle [rad] q0 = [x0, y0, psi_rad0]; % Initial state vector u0 = [v0, delta_rad0]; % Initial control vector fig1 = figure(1); % Figure set-up (fig1) axis([-R_turn R_turn -0*R_turn 2*R_turn]); axis('square') hold on; % Acquire the configuration of robot for plot [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = tricycle_state(q0, u0); plotqb = plot(xb, yb); % Plot vehicle base plotqfw = plot(xfw, yfw, 'r'); % Plot front wheel plotqrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel plotqrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel % Draw fast and erase fast set(gca, 'drawmode','fast'); set(plotqb, 'erasemode', 'xor'); set(plotqfw, 'erasemode', 'xor'); set(plotqrlw, 'erasemode', 'xor'); set(plotqrrw, 'erasemode', 'xor'); % Beginning of simulation for i = 1:n+1 v(i) = v_max*cos(2*delta_rad(i)); u = [v(i), delta_rad(i)]; % Set control input vc = u(1); delta_radc = u(2); to = t(i); tf = t(i)+dt; [t2,q2]=rk4fixed('ks_tricycle_kv',[to tf],q1',2); t1 = t2(2); q1 = q2(2,:); % Acquire the configuration of vehicle for plot [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = tricycle_state(q1, u); % Plot vehicle set(plotqb,'xdata',xb); set(plotqb,'ydata',yb); set(plotqfw,'xdata',xfw); set(plotqfw,'ydata',yfw); set(plotqrlw,'xdata',xrlw); set(plotqrlw,'ydata',yrlw); set(plotqrrw,'xdata',xrrw); set(plotqrrw,'ydata',yrrw); % drawnow pause(0.1); % Pause by 0.2s for slower simulation end q1 = q0; % Set initial state to z1 for simulation

18. Practice Problem A Great Acknowledgement to Prof. R.G. Longoria of Texas University!